In 1997 a groundbreaking technique was introduced by Bobylev [1]}, applied to the hard sphere model in three dimensions with a constant angular function \(b( \hat{u}\cdot \sigma )\) . This new approach proposes that uses the local energy conservation to write the integration of postcollisional molecular velocities in the center of mass-relative velocity framework. That way the integration readily proves a control of the decay rate of such angular averaging for the case of hard spheres in three dimensions with a constant angular cross section.
An important step to obtain evolution estimates for moments is a fundamental lemma that controls the function \(G_{\varphi }(v_*,v)\)  Later, it was extended in [2]} for inelastic collisions. More extension related to the generality of \(b\)  where made later on in [3]}, [4]}. Finally, in [5]}, [6]} the lemma was extended to non cutoff scattering.

